Volume Calculation: Evaluate V = (B * H) / 3 For Given Values

Hey guys! Today, we're diving into a classic volume calculation problem. This is something you'll likely encounter in geometry and various practical applications. We're going to break it down step-by-step, so it's super clear and easy to follow. So, let's get started and figure out how to evaluate the formula! This problem focuses on evaluating a formula, a fundamental skill in mathematics and science. Specifically, we will calculate the volume (V) using the formula V = (B * h) / 3, where B represents the base area and h represents the height. This formula is commonly used to find the volume of pyramids and cones. We are given the values for B as 9 square inches and h as 32 inches. Our goal is to substitute these values into the formula and compute the resulting volume. This exercise reinforces understanding of variable substitution and order of operations in mathematical expressions. Understanding how to manipulate formulas and substitute values is crucial for solving real-world problems. For example, this formula can help determine the volume of a conical container or a pyramid-shaped structure. By mastering this skill, you'll be better equipped to tackle more complex geometric problems and practical applications. So, grab your calculators, and let's get this volume calculated! Remember, math isn't just about numbers; it's about understanding the relationships between them and applying them to the world around us. This problem is a perfect example of how a simple formula can help us understand and quantify three-dimensional space. We will explore the step-by-step process to arrive at the correct solution, highlighting the importance of accurate substitution and arithmetic. Let's dive in!

Understanding the Formula and Given Values

Before we jump into the calculation, let's make sure we're all on the same page with the formula and what each variable represents. Our main task is to evaluate the formula V = (B * h) / 3*. Now, what does this all mean? V stands for volume, which is the amount of space a three-dimensional object occupies. Think of it like how much water you could pour into a container. B represents the base area, which is the area of the bottom surface of the object. In our case, B is given as 9 square inches (in²). This tells us the size of the base. h stands for height, which is the vertical distance from the base to the highest point of the object. We are given h as 32 inches (in). Now that we know what each variable means, we can see that the formula is telling us how to calculate the volume when we know the base area and the height. The formula V = (B * h) / 3 is specifically used for shapes like pyramids and cones. These shapes have a base and taper to a point, which is why we divide by 3 in the formula. If you were calculating the volume of a prism or cylinder (shapes with straight sides), you would use a different formula. So, it's important to recognize the shape you're dealing with to use the correct formula. Now, let's talk about the units. We're given B in square inches (in²) and h in inches (in). When we multiply these together and divide by 3, we'll get the volume in cubic inches (in³). Remember, volume is always measured in cubic units because it represents three-dimensional space. Understanding the units is crucial to ensure your answer makes sense in the context of the problem. If we ended up with square inches for volume, we'd know something went wrong! So, to recap, we have a formula that relates volume to base area and height, and we have the values for B and h. The next step is to substitute these values into the formula and do the math. We're well on our way to finding the volume! Keep these concepts in mind as we move forward; a strong understanding of the fundamentals will make the calculation much easier.

Step-by-Step Calculation

Alright, guys, now comes the fun part – plugging in the numbers and doing the calculation! This is where we put our understanding of the formula into action. Remember, our formula is V = (B * h) / 3*. We know that B = 9 in² and h = 32 in. The first step is to substitute these values into the formula. This means replacing the variables B and h with their respective numbers. So, our equation now looks like this: V = (9 in² * 32 in) / 3. See? We've just swapped the letters for the numbers. Now, we need to follow the order of operations, which you might remember as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In this case, we have multiplication and division. According to PEMDAS, we perform multiplication before division. So, let's multiply 9 in² by 32 in. If you punch that into your calculator (or do it by hand!), you'll find that 9 * 32 = 288. So, now our equation looks like this: V = 288 in³ / 3. Notice that when we multiplied square inches (in²) by inches (in), we got cubic inches (in³). This is because we're now dealing with a three-dimensional measurement (volume). The next step is to divide 288 in³ by 3. This is the final step in our calculation. 288 divided by 3 is 96. So, V = 96 in³. And that's it! We've calculated the volume. It's always a good idea to double-check your work, especially in math. Make sure you've substituted the values correctly and performed the arithmetic accurately. A small mistake in the calculation can lead to a wrong answer. But in this case, we've followed the steps carefully, so we can be confident in our result. So, our final answer is 96 cubic inches. This means the volume of the object (whether it's a pyramid or a cone) is 96 cubic inches. We've successfully evaluated the formula for the given values. Awesome job, guys! Now, let's see which of the answer choices matches our result.

Identifying the Correct Answer

Okay, we've done the hard work of calculating the volume. Now it's time to match our result with the given options. We found that the volume, V, is equal to 96 cubic inches (96 in³). Let's look at the answer choices provided: A. 288 in³ B. 32 in³ C. 9.6 in³ D. 96 in³ Which one matches our answer? If you guessed D, you're absolutely right! Option D, 96 in³, is the same as our calculated volume. This confirms that we've gone through the steps correctly and arrived at the correct solution. It's always satisfying when your calculations match one of the answer choices. But even if none of the options matched your result, it wouldn't necessarily mean you're wrong. It could be a mistake in the answer choices themselves. In such cases, it's important to double-check your work carefully and trust your calculations. But in this case, we have a match, so we can be confident in our answer. Let's quickly look at the other options to see why they're incorrect. Option A, 288 in³, is the result of multiplying 9 and 32 but forgetting to divide by 3. Option B, 32 in³, might be a result of only considering the height and neglecting the base area and the division by 3. Option C, 9.6 in³, seems like a random number and doesn't follow logically from the formula and given values. So, by comparing our calculated answer with the options and understanding the potential errors that could lead to the incorrect choices, we can be even more confident that we've selected the right answer. Choosing the correct answer isn't just about finding a matching number; it's about understanding the entire process and why that number is the solution. We've not only found the correct answer, but we've also reinforced our understanding of volume calculation. That's a win-win! Great job, everyone! We've successfully navigated this problem from start to finish.

Key Takeaways and Practice Problems

Alright, guys, we've conquered this volume calculation problem! Before we wrap up, let's recap the key things we learned and see how we can apply them to other situations. The most important takeaway here is understanding how to evaluate a formula. We started with the formula V = (B * h) / 3, and we were given values for B and h. The key was to substitute those values correctly into the formula and then perform the calculations following the order of operations. This skill of substitution and evaluation is fundamental in math and science. You'll use it in countless situations, from simple algebra problems to complex physics equations. Another crucial aspect was understanding the units. We were given B in square inches and h in inches, and we ended up with a volume in cubic inches. Always pay attention to the units, as they can help you catch mistakes and ensure your answer makes sense. Remember, volume is a three-dimensional measurement, so it's always expressed in cubic units. We also reinforced the importance of the order of operations (PEMDAS). Multiplying before dividing in this case was essential to getting the correct answer. A simple mistake in the order of operations can throw off your entire calculation. So, always keep PEMDAS in mind! Finally, we learned how to connect the formula to a real-world concept. The formula V = (B * h) / 3 is used to calculate the volume of shapes like pyramids and cones. Understanding this connection helps you visualize the problem and apply the formula appropriately. To solidify your understanding, let's think about some practice problems. What if B was 12 in² and h was 20 in? What would the volume be? Or, what if you were given the volume and the base area, and you needed to find the height? How would you rearrange the formula to solve for h? These types of practice problems will help you master the formula and become more confident in your problem-solving abilities. Remember, math is like a muscle – the more you use it, the stronger it gets! So, keep practicing, and you'll become a volume calculation pro in no time. We've covered a lot today, from understanding the formula to identifying the correct answer. Keep these key takeaways in mind, and you'll be well-prepared for similar problems in the future. You got this!